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A property of Dunford-Pettis type in topological groups

Authors: E. Martín-Peinador and V. Tarieladze
Journal: Proc. Amer. Math. Soc. 132 (2004), 1827-1837
MSC (2000): Primary 22A05; Secondary 46A16
Published electronically: October 21, 2003
MathSciNet review: 2051147
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Abstract: The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

References [Enhancements On Off] (What's this?)

  • 1. Aussenhofer, L.,
    Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups.
    Dissertationes Mathematicae 384 (1999), 113 pp. MR 2001c:43003
  • 2. Banaszczyk, W.,
    Additive Subgroups of Topological Vector Spaces,
    Lecture Notes in Mathematics 1466,
    Springer-Verlag, Berlin, Heidelberg, New York, 1991. MR 93b:46005
  • 3. Banaszczyk, W., and E. Martín-Peinador,
    The Glicksberg theorem on weakly compact sets for nuclear groups,
    Annals of the New York Academy of Sciences 788 (1996), 34-39. MR 99e:22001
  • 4. Bombal, F., and I. Villanueva,
    On the Dunford-Pettis property of the tensor products of $C(K)$ spaces,
    Proc. Amer. Math. Soc. 129 (2001), 1359-1363. MR 2001h:46024
  • 5. Bruguera, M.,
    Grupos topológicos y grupos de convergencia: estudio de la dualidad de Pontryagin,
    Doctoral Dissertation, 1999.
  • 6. Cascales, B., and J. Orihuela,
    On compactness in locally convex spaces,
    Math. Z. 195 (1987), 365-381. MR 88i:46021
  • 7. Cembranos, P.,
    The hereditary Dunford-Pettis property on $C(K,E)$,
    Illinois J. Math. 31 (1987), 365-373. MR 88g:46028
  • 8. Cembranos, P.,
    The hereditary Dunford-Pettis property for $l_1(E)$,
    Proc. Amer. Math. Soc. 108 (1990), 947-950. MR 90i:46019
  • 9. Chasco, M. J.,
    Pontryagin duality for metrizable groups,
    Archiv der Math. 70 (1998), 22-28. MR 98k:22003
  • 10. Chasco, M. J., E. Martín-Peinador, and V. Tarieladze,
    On Mackey topology for groups,
    Studia Math. 132 (1999), 257-284. MR 2000b:46003
  • 11. Comfort, W. W., S. Hernández, and F. J. Trigos-Arrieta,
    Relating a locally compact Abelian group to its Bohr compactification,
    Advances in Math. 120 (1996), 322-344. MR 97k:22005
  • 12. Comfort, W. W., and K. A. Ross,
    Topologies induced by groups of characters,
    Fundamenta Math. 55 (1964), 283-291. MR 30:183
  • 13. Diestel, J.,
    A survey of results related to the Dunford-Pettis property,
    Contemporary Mathematics 2 (1980), 15-60. MR 82i:46023
  • 14. Diestel, J.,
    Sequences and Series in Banach Spaces,
    Graduate Texts in Mathematics 92,
    Springer-Verlag, New York, 1984. MR 85i:46020
  • 15. Dunford, N., and B. J. Pettis,
    Linear operations on summable functions,
    Trans. Amer. Math. Soc. 47 (1940), 323-392. MR 1:338b
  • 16. Galindo, J., S. Hernández, and S. Macario,
    A characterization of the Schur property by means of the Bohr topology,
    Topology and Appl. 97 (1999), 99-108. MR 2000k:22002
  • 17. Glicksberg, I.,
    Uniform boundedness for groups,
    Canadian Journal of Math. 14 (1962), 269-276. MR 27:5856
  • 18. Grothendieck, A.,
    Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$,
    Canad. J. Math. 5 (1953), 129-173. MR 15:438b
  • 19. Hewitt, E., and K. A. Ross,
    Abstract Harmonic Analysis I,
    Die Grundlehren der Mathematischen Wissenschaften 115, Springer-Verlag, Berlin, 1963. MR 28:158
  • 20. Martín-Peinador, E.,
    A reflexive admissible topological group must be locally compact,
    Proc. Amer. Math. Soc. 123 (1995), 3563-3566. MR 96a:22002
  • 21. Remus, D., and F. J. Trigos-Arrieta,
    Abelian groups which satisfy Pontryagin duality need not respect compactness,
    Proc. Amer. Math. Soc. 117 (4) (1993), 1195-1200. MR 93e:22009
  • 22. Smith, M. F.,
    The Pontryagin duality theorem in linear spaces,
    Ann. Math. 56 No. 2 (1952), 248-253. MR 14:183a
  • 23. Varopoulos, N. T.,
    Studies in harmonic analysis,
    Proc. Cambridge Philos. Soc. 60 (1964), 465-516. MR 29:1284

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Additional Information

E. Martín-Peinador
Affiliation: Departamento de Geometría y Topología, Universidad Complutense de Madrid, Av. Complutense s/n, Madrid 28040, España

V. Tarieladze
Affiliation: Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, Akuri str. 8. Tbilisi 93, Georgia

Keywords: Dunford-Pettis property, Schur property, Bohr topology, dual group, Pontryagin reflexive, locally convex space.
Received by editor(s): June 26, 2001
Received by editor(s) in revised form: April 20, 2002, and January 28, 2003
Published electronically: October 21, 2003
Additional Notes: This paper was written while the second author was enjoying a Sabático at the Complutense University of Madrid. The first author was partially supported by D.G.I.C.Y.T. BFM 2000-0804-C03-01.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

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